For general nonlinear control systems, we present a novel approach to adaptive control, which employs a certainty-equivalence (indirect) control law and an identifier with event-triggered updates of the plant parameter estimates, where the triggers are based on the size of the plant's state, and the updates are conducted using a nonrecursive least-squares estimation over certain finite time intervals, with updates employing delayed measurements of the state. With a suitable nonrestrictive parameter-observability assumption, our adaptive controller guarantees global stability, regulation of the plant state, and our identifier achieves parameter convergence, in finite time, even in the absence of persistent excitation, for all initial conditions other than those where the initial plant state is zero. The robustness of our event-triggered adaptive control scheme to vanishing and nonvanishing disturbances is verified in simulations with the assistance of a dead-zone-like modification of the update law. The major distinctions of our approach from supervisory adaptive schemes are that our approach is indirect and our triggering is related to the control objective (the regulation error). The major distinction from the classical indirect Lyapunov adaptive schemes based on tuning related to the regulation error is that our approach does not involve a complex redesign of the controller to compensate for the detrimental effects of rapid tuning on the transients by incorporating the update law into the control law. Instead, our approach allows for the first time to use a simple certainty-equivalence adaptive controller for general nonlinear systems.